486 
ME. E. W. BARXES OX IXTEGEAL FUXCTIOXS. 
We have 
lo g F 0 
- X (9o) = £(— o"- 
s ii~ 1 r m rj' 
X t n a log ;/ -j- | t n° log n dn — - n <r log /< + ..■= r V, l cr), 
«=i 
— F, = — £ (— t S + cr). 
»TT* 
And, as in § 77, f(z) = 
+ 1 
. 7r (cr + 1) (cr A 1) 
Sill- 
Thus the asymptotic expansion of F ( 2 ) may be written 
-A(-<r) e sin-(?Ai) 
<r + l 
(-)- ] ?(-<T- rs) 
8= —p $ Z 
Note that 
l (0) # = -i, t (0)f = - l log 277. 
1 
Hence, when cr = 0, r = -, we get the asymptotic expansion 
n 
n =1 
L + 1 <2 
MP 
ii=l m V n 
— Z * e sin - p ' 2p log 2 7T + 
(-)" H( p) 
*=~P 
which agrees with the expansion of § 65. 
Simple Repeated Functions of Finite Integral Order. 
§81. It is obvious from the investigations of §§ 70-73 that the asymptotic expansion 
, where p < p < p + 1 and p is such 
that X is convergent, and X divergent, will hold in the limit when p — p, 
-■n G>n 
provided that in any terms which become infinite we reject the infinite part and 
keep only the corresponding finite expression found by applying the usual methods of 
the calculus of limits to the subsidiary Fourier and Maclaurin series. Consider, for 
example, the function 
obtained in § 70 for II 
n= 1 
/ 2 \P» V 1 / r. \„ 
1 + - 
\ ] 
II 
,' = 1 
where 
(7-4-1 
The asymptotic expansion obtained previously was 
* “ Theory of the Gamma Function,” § 27, 
t Ibid., § 30. 
